Math 2250 Lab 10
Name/Unid:
These problems are w10.2, w10.3, and w10.4 respectively in the homework that was originally due on Friday March 21, but that is now due on Monday March 24. There are no
separate lab problems this week (no ”part 2”).
Consider a mass-spring-dashpot system with an additional external force F (t) being applied to the mass. In particular we consider a periodic external force F (t) = F0 cos(!t). As
we know, this system is governed by the following di↵erential equation for the displacement
x(t) from equilibrium:
m · x00 (t) + c · x0 (t) + k · x(t) = F0 cos(!t)
Take m =1 kg, k =0.64 N/m, and F0 = 2 N. Also the system will always start at rest, i.e.
x(0) = 0, x0 (0) = 0 The damping coefficient c will be modified in di↵erent problems, as will
the angular frequency ! of the driving force.
1. Consider the configuration above, in the undamped case c = 0. In particular consider
the initial value problem
x00 (t) + 0.64 · x(t) = 2cos(!t)
x(0) = 0
x0 (0) = 0
(a) What is the natural angular frequency !0 (for the unforced problem) in this di↵erential equation? Hint: the natural frequency is defined to be the angular frequency
for the solutions to the unforced and undamped di↵erential equation, which in this
case is the DE
x00 (t) + 0.64 · x(t) = 0.
(b) Assume that ! 6= !0 . Use the method of undetermined coefficients to solve for
the particular solution xP (t). Then use x(t) = xP (t) + xH (t) to solve the initial
value problem for x(t), where xH (t) is a solution to the corresponding homogeneous
(unforced) equation. Use a technology to check that this solution is correct.
(c) Write down the specific case of the solution x(t) (found in part b) for ! = 0.7.
Compute the period of this solution, which is a superposition of two cosine functions.
Use technology to graph one period of the solution. What phenomenon is somewhat
exhibited by this solution?
(d) Solve the IVP when ! = !0 . Use the method of undetermined coefficients to solve
for a new particular solution, then use x(t) = xP (t) + xH (t) to solve the initial value
problem for x(t). Graph this solution on the interval 0  t  60 seconds. What
phenomenon is exhibited by this solution?
2. Consider the same mass-spring-dashpot system as above, except with c = 2 kg/s, and
! = 0.8. This gives us the di↵erential equation
x00 (t) + 2x0 (t) + 0.64x(t) = 2cos(0.8 · t)
(a) Use the method of undetermined coefficients to find a particular solution xP (t) to
this di↵erential equation.
(b) Use the particular solution xP (t) found in part (a) and the solution xH (t) to the
corresponding homogeneous equation to write down the general solution to this
di↵erential equation. Identify the ”steady periodic” and ”transient” parts of this
general solution.
(c) Solve the initial value problem for this di↵erential equation above, with x(0) = 0,
x0 (0) = 0. (You may use technology to do this, which will also check your work in
parts a,b.)
(d) Graph, on a single plot, the steady periodic solution from (b) and the solution to
the IVP in (c). Choose a time interval so that you can clearly see the convergence
of the IVP solution to the steady periodic solution.
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3. Consider the same forced oscillator equation, except with a relatively small damping
coefficient of c = 0.2 Unlike in the previous problem we will consider variable angular
frequencies !. This gives following di↵erential equation:
x00 (t) + 0.2x0 (t) + 0.64x(t) = 2cos(!t)
This system is a slightly damped forced oscillator, and we will investigate its practical
resonance:
(a) Use the formula (21) in section 5.6, on page 384 of the textbook (page 355 in older
version) to create a plot of the amplitude C of the steady periodic solution xsp (t)
as a function of the driving angular frequency !.
(b) Explain why the steady periodic amplitude peaks at a value near ! = 0.8. Use
calculus to find the exact value of ! that gives the maximum amplitude C.
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